Selina Solutions Concise Maths Class 7 Chapter 14: Lines and Angles (Including Construction of Angles) Exercise 14B

Selina Solutions Concise Maths Class 7 Chapter 14 Lines and Angles (Including Construction of Angles) Exercise 14B contains 100% accurate answers, strictly based on the ICSE syllabus. The concept of a transversal, parallel lines and the conditions of parallelism are the topics which are covered under this exercise. The important shortcut tricks and formulas are explained in simple language to impart better understanding among students. Selina Solutions Concise Maths Class 7 Chapter 14 Lines and Angles (Including Construction of Angles) Exercise 14B, PDF can be downloaded from the links available below.

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Exercise 14B page: 166

1. In questions 1 and 2, given below, identify the given pairs of angles as corresponding angles, interior alternate angles, exterior alternate angles, adjacent angles, vertically opposite angles or allied angles:

(i) ∠3 and ∠6
(ii) ∠2 and ∠4
(iii) ∠3 and ∠7
(iv) ∠2 and ∠7
(v) ∠4 and∠6
(vi) ∠1 and ∠8
(vii) ∠1 and ∠5
(viii) ∠1 and ∠4
(ix) ∠5 and ∠7

Solution:

(i) ∠3 and ∠6 are interior alternate angles.

(ii) ∠2 and ∠4 are adjacent angles.

(iii) ∠3 and ∠7 are corresponding angles.

(iv) ∠2 and ∠7 are exterior alternate angles.

(v) ∠4 and∠6 are allied or co-interior angles.

(vi) ∠1 and ∠8 are exterior alternate angles.

(vii) ∠1 and ∠5 are corresponding angles.

(viii) ∠1 and ∠4 are vertically opposite angles.

(ix) ∠5 and ∠7 are adjacent angles.

2. (i) ∠1 and ∠4
(ii) ∠4 and ∠7
(iii) ∠10 and ∠12
(iv) ∠7 and ∠13
(v) ∠6 and ∠8
(vi) ∠11 and ∠8
(vii) ∠7 and ∠9
(viii) ∠4 and ∠5
(ix) ∠4 and ∠6
(x) ∠6 and ∠7
(xi) ∠2 and ∠13

Solution:

(i) ∠1 and ∠4 are vertically opposite angles.

(ii) ∠4 and ∠7 are interior alternate angles.

(iii) ∠10 and ∠12 are vertically opposite angles.

(iv) ∠7 and ∠13 are corresponding angles.

(v) ∠6 and ∠8 are vertically opposite angles.

(vi) ∠11 and ∠8 are allied or co-interior angles.

(vii) ∠7 and ∠9 are vertically opposite angles.

(viii) ∠4 and ∠5 are adjacent angles.

(ix) ∠4 and ∠6 are allied or co-interior angles.

(x) ∠6 and ∠7 are adjacent angles.

(xi) ∠2 and ∠13 are allied or co-interior angles.

3. In the following figures, the arrows indicate parallel lines. State which angles are equal. Give reasons.

Solution:

(i) From the figure (i)

a = b are corresponding angles

b = c are vertically opposite angles

a = c are alternate angles

So we get

a = b = c

(ii) From the figure (ii)

x = y are vertically opposite angles

y = l are alternate angles

x = l are corresponding angles

1 = n are vertically opposite angles

n = r are corresponding angles

So we get

x = y = l = n = r

Similarly

m = k are vertically opposite angles

k = q are corresponding angles

Hence, m = k = q.

4. In the given figure, find the measure of the unknown angles:

Solution:

From the figure

a = d are vertically opposite angles

d = f are corresponding angles

f = 110⁰ are vertically opposite angles

So we get

a = d = f = 110⁰

We know that

e + 110⁰ = 180⁰ are linear pair of angles

e = 180 – 110 = 70⁰

b = c are vertically opposite angles

b = e are corresponding angles

e = g are vertically opposite angles

So we get

b = c = e = g = 70⁰

Therefore, a = 110⁰, b = 70⁰, c = 70⁰, d = 110⁰, e = 70⁰, f = 110⁰ and g = 70⁰.

5. Which pair of the dotted line, segments, in the following figures, are parallel. Give reason:

Solution:

(i) From the figure (i)

If the lines are parallel we get 120⁰ + 50⁰ = 180⁰

There are co-interior angles where 170⁰ ≠ 180⁰

Therefore, they are not parallel lines.

(ii) From the figure (ii)

∠1 = 45⁰ are vertically opposite angles

We know that the lines are parallel if

∠1 + 135⁰ = 180⁰ are co-interior angles

Substituting the values

45⁰ + 135⁰ = 180⁰

180⁰ = 180⁰ which is true

Therefore, the lines are parallel.

(iii) From the figure (iii)

The lines are parallel if corresponding angles are equal

Here 120⁰ ≠ 130⁰

Hence, lines are not parallel.

(iv) ∠1 = 110⁰ are vertically opposite angles

We know that if lines are parallel

∠1 + 70⁰ = 180⁰ are co-interior angles

Substituting the values

110⁰ + 70⁰ = 180⁰

180⁰ = 180⁰ which is correct

Therefore, the lines are parallel.

(v) ∠1 + 100⁰ = 180⁰

So we get

∠1 = 180⁰ – 100⁰ = 80⁰ which is a linear pair
Here the lines l₁ and l₂ are parallel if ∠1 = 70⁰
But here ∠1 = 80⁰ which is not equal to 70⁰
So the lines l₁ and l₂ are not parallel
Now, l₃ and l₅ will be parallel if corresponding angles are equal
But here, the corresponding angles 80⁰ ≠ 70⁰
Hence, l₃ and l₅ are not parallel.
We know that, in l₂ and l₄
∠1 = 80⁰ are alternate angles
80⁰ = 80⁰ which is true
Hence, l₂ and l₄ are parallel.

(vi) Two lines are parallel if alternate angles are equal

Here, 50⁰ ≠ 40⁰ which is not true

Hence, the lines are not parallel.

6. In the given figures, the directed lines are parallel to each other. Find the unknown angles.

Solution:

(i) If the lines are parallel

a = b are corresponding angles

a = c are vertically opposite angles

a = b = c

Here b = 60⁰ are vertically opposite angles

Therefore, a = b = c = 60⁰

(ii) If the lines are parallel

x = z are corresponding angles

z + y = 180⁰ is a linear pair

y = 55⁰ are vertically opposite angles

Substituting the values

z + 55⁰ = 180⁰

z = 180 – 55 = 125⁰

If x = z we get x = 125⁰

Therefore, x = 125⁰, y = 55⁰ and z = 125⁰.

(iii) If the lines are parallel

c = 120⁰ (corresponding angles)

a + 120⁰ = 180⁰ are co-interior angles

a = 180 – 120 = 60⁰

We know that a = b are vertically opposite angles

So b = 60⁰

Therefore, a = b = 60⁰ and c = 120⁰.

(iv) If the lines are parallel

x = 50⁰ are alternate angles

y + 120⁰ = 180⁰ are co-interior angles

y = 180 – 120 = 60⁰

We know that

x + y + z = 360⁰ are angles at a point

Substituting the values

50 + 60 + z = 360

By further calculation

110 + z = 360

z = 360 – 110 = 250⁰

Therefore, x = 50⁰, y = 60⁰ and z = 250⁰.

(v) If the lines are parallel

x + 90⁰ = 180⁰ are co-interior angles

x = 180⁰ – 90⁰ = 90⁰

∠2 = x

∠2 = 90⁰

We know that the sum of angles of a triangle

∠1 + ∠2 + 30⁰ = 180⁰

Substituting the values

∠1 + 90⁰ + 30⁰ = 180⁰

By further calculation

∠1 + 120⁰ = 180⁰

∠1 = 180 – 120 = 60⁰

Here ∠1 = k are vertically opposite angles

k = 60⁰

Here ∠1 = z are corresponding angles

z = 60⁰

Here k + y = 180⁰ are co-interior angles

Substituting the values

60⁰ + y = 180⁰

y = 180 – 60 = 120⁰

Therefore, x = 90⁰, y = 120⁰, z = 60⁰, k = 60⁰.

7. Find x, y and p is the given figures:

Solution:

(i) From the figure (i)

The lines are parallel

x = z are corresponding angles

y = 40⁰ are corresponding angles

We know that

x + 40⁰ + 270⁰ = 360⁰ are the angles at a point

So we get

x + 310⁰ = 360⁰

x = 360 – 310 = 50⁰

So z = x = 50⁰

Here p + z = 180⁰ is a linear pair

By substituting the values

p + 50⁰ = 180⁰

p = 180 – 50 = 130⁰

Therefore, x = 50⁰, y = 40⁰, z = 50⁰ and p = 130⁰.

(ii) From the figure (ii)

The lines are parallel

y = 110⁰ are corresponding angles

We know that

25⁰ + p + 110⁰ = 180⁰ are angles on a line

p + 135⁰ = 180⁰

p = 180 – 135 = 45⁰

We know that the sum of angles of a triangle

x + y + 25⁰ = 180⁰

x + 110⁰ + 25⁰ = 180⁰

By further calculation

x + 135⁰ = 180⁰

x = 180 – 135 = 45⁰

Therefore, x = 45⁰, y = 110⁰ and p = 45⁰.

8. Find x in the following cases:

Solution:

(i) From the figure (i)

The lines are parallel

2x + x = 180⁰ are co-interior angles

3x = 180⁰

x = 180/3 = 60⁰

(ii) From the figure (ii)

The lines are parallel

4x + ∠1 = 180⁰ are co-interior angles

∠1 = 5x are vertically opposite angles

Substituting the values

4x + 5x = 180⁰

So we get

9x = 180⁰

x = 180/9 = 20⁰

(iii) From the figure (iii)

The lines are parallel

∠1 + 4x = 180⁰ are co-interior angles

∠1 = x are vertically opposite angles

Substituting the values

x + 4x = 180⁰

5x = 180⁰

So we get

x = 180/5 = 36⁰

(iv) From the figure (iv)

The lines are parallel

2x + 5 + 3x + 55 = 180⁰ are co-interior angles

5x + 60⁰ = 180⁰

By further calculation

5x = 180 – 60 = 120⁰

So we get

x = 120/5 = 24⁰

(v) From the figure (v)

The lines are parallel

∠1 = 2x + 20⁰ are alternate angles

∠1 + 3x + 25⁰ = 180⁰ is a linear pair

Substituting the values

2x + 20⁰ + 3x + 25⁰ = 180⁰

5x + 45⁰ = 180⁰

So we get

5x = 180 – 45 = 135⁰

x = 135/5 = 27⁰

(vi) From the figure (vi)

Construct a line parallel to the given parallel lines

∠1 = 4x and ∠2 = 6x are corresponding angles

∠1 + ∠2 = 130⁰

Substituting the values

4x + 6x = 130⁰

10x = 130⁰

So we get

x = 130/10 = 13⁰